Igor Rumanov, Department of Applied Mathematics, University of Colorado Boulder
(2+1)-dimensional Whitham systems: 2dNLS and KP vs. ‘hydrodynamic’ systems in one and two spatial dimensions
The main result of this talk is the recently obtained Whitham modulation system for the (2+1)-dimensional nonlinear Schroedinger equation (2dNLS) (joint work with M. J. Ablowitz and J. T. Cole). The first applications demonstrating its validity are the linear stability analysis of plane periodic traveling waves with its help and its KP Whitham limit. The need for finding solutions of such systems raises a number of interesting questions.
I will review the 2dNLS and KP Whitham systems in the context of the theory of previously known hydrodynamic integrable systems solvable by the generalized hodograph method. Understanding the 2dNLS Whitham system may be achieved by studying its multiple interesting reductions including the KP Whitham system. While the KP Whitham system is simpler and supposed to be integrable, finding its general solutions is still a challenge. The theory of hydrodynamic reductions may help as well as a better understanding of the well-known (1+1)-dimensional Whitham systems.